Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have

\begin{equation} \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ \chi_{5,2}&=(1, i, -i, -1, 0),\qquad\qquad \text{(1)}\\ \chi_{5,3}&=(1, -1, -1, 1, 0),\\ \chi_{5,4}&=(1, -i, i, -1, 0),\\ \end{aligned} \end{equation}

We construct a sum involving Dirichlet character $\chi_{q}$ as

\begin{equation} \begin{aligned} Q(\chi_{q})&=\sum_{k=1}^{q-1}k\chi_q(k)\qquad\qquad \text{(2)} \end{aligned} \end{equation}

**Proposition A:**
For a complex $\chi_q$, like $\chi_{5,2},\chi_{5,4}$,
\begin{equation}
\begin{aligned}
&\color{red}{\text{sign}(\mathrm{Re}\chi_{q}(-1))}Q(\mathrm{Re}\chi_{q})>0,\qquad\text{(3)}\\
&\color{red}{\text{sign}(\mathrm{Im}\chi_{q}(-2))}Q(\mathrm{Im}\chi_{q})>0.\qquad\text{(4)}\\
\end{aligned}
\end{equation}

We are seeking a proof or a reference of the proof for this proposition.

This problem is related to a partial answer of another problem titled sign unchanged for Dirichlet polynomials? that we posted earlier at math.stackexchange.com. The motivation of studying this problem is also mentioned there.